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Full text of "Treatise On Analysis Vol-Ii"

4       XII   TOPOLOGY AND TOPOLOG1CAL ALGEBRA

The equivalence of (a) and (b) follows from the criterion of continuity
(3.11.4(b)), and the equivalence of (b) and (c) from the definition of a con-
tinuous function.

Remark

(12.2.2)    Let (Ua)a, be an open covering (3.16) of a topological space E. For
a set G c E to be open in E it is necessary and sufficient that each of the sets
G n Ua should be open in the subspace Ua. This follows immediately from the
axioms (Oj) and (On) and the relation G = (J (G n Ua). Taking complements,

ael

it follows that a set F c E is closed m E if and only if each of the sets F n Ua is
closed in the subspace Ua.

(12.2.3)    Let L be a subset of a topological space E. Then the following proper-
ties are equivalent:

(a)    For each xelu there exists a neighborhood V of x in E such that
L n V is closed in V;

(b)    L is an open subset of the subspace L (the closure ofL in E);

(c)    L is the intersection of an open subset and a closed subset of'E.

It is clear that (b) implies (c), L being the intersection of L with an open
subset of E. Equally clearly, (c) implies (a). Let us show that (a) implies (b).
For each x e L, we have V n L = V n L, because V n L is closed in V; this
shows that in the subspace L the point x is an interior point of L, and therefore
L is open in L.

When L satisfies the equivalent conditions of (12.2.3), it is said to be a
locally closed subset of E.

(12.2.4)    A procedure for constructing topological spaces which is frequently
used is that of "patching together" a family of topological spaces (EA)AeL in
such a way that, in the topological space E so obtained, the EA are identified
with open sets of E. Since pairs of these sets may very well intersect, this identi-
fication requires that, for each pair of indices (1, u), we are given a homeomor-
phism of an open subset of EA onto an open subset of EM.

To be precise, suppose that for each pair (1, /^) e L x L we are given:

(1)    an open subset AA/[t of EA;

(2)    a homeomorphism h^: AA/i -* A^,

satisfying the following conditions:efinitions and results of (3.19) relating to